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## Topology

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**Topology**• What is a polygon? • Set operations • Interior, boundary, exterior • Skin, Hair, Wound, Cut, and regularization • Components, holes • Polygons and faces • Loops • A linear geometric complex**Motivation**• We need a terminology and notation to describe the domain for any particular representation or operation and the precise nature of the result. • How would you distinguish these situations?**Symbols and notation**• ! complement, union, intersection, XOR We often use + for union and no operator for intersection ! has highest priority, then intersection: !A(!B+CD) • inclusion, = equality • member of, not member of • empty set, Ω whole space (Euclidean) • Set definition p: pA and pB} • each, there is, implies, iff • AB≠ sets interfere, AB=, AC=… exclusive • a neighborhood (infinitely small ball) around a point**List the letters corresponding to drawings that represent**valid polygons according to your definition. B C D E F G A Define a polygon Write the definition of “polygon” on a sheet of paper with your name on it.**Set theoretic operations**complement union !S = {s: s S} AB = {s: s A or s B} also written A+B ! has highest priority deMorgan Laws !!A=A !(A+B) = !A!B !(AB) = !A+!B intersection AB = {s: s A and s B} also written AB**Differences**A A B B AB = A!B+!AB A\B = A!B called XOR and also symmetric difference also written A–B**Inclusion**A B A B complete the equivalence (A B) ( , pA pB) A=B AB and BA**A set is bounded (or equivalently finite) if it is contained**in a ball of finite radius. A line, a ray are not bounded A disk, and edge, are Bounded (finite)**Boundary of 2D point-sets**a point and its neighborhood (tiny ball around it) point in the interior iS of S (surrounded by disk entirely in S) S.e S.i point in the exterior eS of S (surrounded by disk entirely out of S) S S.b point on the boundary bS of S (all surrounding disks intersect S and !S) the point is touching S and !S**How to draw the boundary**Although the boundary has 0 thickness, we draw it as a thick line S S.b**Interior, boundary, exterior**S.e S.b S.i**Open set**A set is said to be open when it does not include any of its boundary S=S.i contains no points that touch S.b**Closed set**A set is closed if it contains its boundary S=S.i+S.b S.b S**Interior operator**• The interior operator returns the difference between the set and its boundary S.i=S–S.b interior S**Closure operator**The closure operator returns the union of the set with its boundary S.k=S+S.b closure S**Drawing the boundary**• Use color or fill to indicate which portions of the boundary are included in the set S. vertex included in S edge included in S (does not include its end points) vertex not included**Drawing interior boundaries**• Interior boundaries = boundary portions not touching S.e vertices not in S P Must be drawn (not filled) to indicate that they are not part of S edge not in S rest is omitted (not drawn) for simplicity vertex included in S drawn filled**Interior, boundary, and closure**S out cells are omitted when obvious S.k S.i S.b**Regularization operator**• The regularization S.r of set S is S.i.k, the closure of its interior S closed-regularization We can also define the open-regularization S.k.i interior S.i.k S.i closure**Decomposing the boundary**Membrane S.m separates S.i from S.e Cut Skin Interior Hair Exterior Wound Any set S defines a decomposition of Ω into 6 exhaustive & exclusive sets: S.i, S.e, S.s, S.w, S.h, S.c**Definitions**• Boundary: S.b = points touching to S and !S • Interior: S.i = S – S.b • Exterior: S.e = !S – S.b • Membrane: S.m = S.i.bS.e.b • Skin: S.s = S.mS • Wound: S.w = S.m–S • Dangle: S.d = S.b–S.m • Hair: S.h = S.dS • Cut: S.c = S.d–S**Regularized sets**A set is closed-regularized if it is equal to the closure of its interior S = S.i.k A set is open-regularized if it is equal to the interior of its closure S = S.k.i Regularized sets have no cut and no hair**A set S is connected if from every point p in S one can walk**to every other point q in S along a curve C that lies entirely in S. A non-empty set S has one or more maximally connected components, which we will call the components of S If a connected subset of S contains a component U, then S=U. Connected components Draw a set S that is not connected but whose closure S.k is.**vertex not in S**Connected components This set is closed and connected Can join any pair of points by a curve in S This open set is not connected: it has 2 components**A hole in S is a bounded components of its complement**For example a closed cavity in a 3D shape Do not confuse a hole with the concave part where the coffee stays or with the through-hole in the handle of a mug Holes of a bounded set NOT a hole in a 3D set. S has genus 1 (1 handle) Hole in a 2D set Cannot say where the handle is!**Closed set with one hole**Examples of holes Simply connected closed set Open set with no hole Closed set with one hole**Manifold boundary**Connected set with a non-manifold boundary Connected set with a manifold boundary in 2D removing a non-manifold vertex would change the the number of connected components or of holes (which are the connected components of the complement)**Write the definition of a polygon**2-cell (region), not its bounding curve connected may have holes needs not be manifold is bounded by straight line segments (a finite number) is bounded (finite) Do not confuse a topological definition with a particular data structure or representation scheme**Definition of a polygon**Connected, bounded, open-regularized, subset of the plane, with boundary is a subset of a finite union of lines. A polygon is open: does not contain its boundary A polygon is regularized: has no cut (S=S.k.i) A polygon can be non-manifold A polygon may have holes**Cells of a polygon**• Given a polygon, do we know what are its edges and vertices? • Note that this is different from the question: given a set of edges and vertices, what is the polygon they define. • Because a representation of a polygon in terms of edges and vertices may be • ambiguous • invalid • We will discuss a representation scheme for modeling polygons**Vertices and edges of a polygon**Crossings Non-manifold vertices Vertices Non-smooth points of S.b that are not crossings Connected components of the boundary without vertices and crossings Edges Edges, vertices, crossings are exclusive**Circuits of a polygon**Circuit 2 The circuits of S are the components of S.b Circuits are exclusive Circuit 1 Can they be represented as a circular list of vertex ids? The same vertex may appear more than once, but the circuit should not cross itself**Computing the circuits**Edges = roads Vertices = turns Crossings = intersections Each tour visits a loop Pick unvisited edge Walk along interior sidewalk - keep road on your left - never cross a street Each circuit may be represented by a circular list of vertex ids (with replications). S.b is the union of all its circuits (which are disjoint)**Definition of a Face**Open, connected, subset of the plane, with boundary in the union of a finite number of lines Needs not be regularized: may have a cut Needs not be bounded: may be infinite**Want polyhedra boundaries to be decomposed into an exclusive**set of vertices, edges, and faces Polyhedra are bounded by faces, not by polygons Why distinguish faces & polygons**Faces from edges of arrangement**• Consider the union U of edges in the plane Ω • The connected components of Ω–U are faces The edges define an arrangement The faces are the 2-cells of the arrangement A (unbounded face) C B**How to compute the arrangement**• Split all edges are their intersections and at their intersections with vertices of other edges • Build circuits for each face • Keep edges to your left and never cross them • Build an inclusion tree of circuits • Assume a sidewalk at infinity around everything • Each node is a loop that contains all its children • Cast ray inwards and use parity of # of intersections • Identify faces and their boundaries • Nodes at even graph-distance from the root are the outer boundaries of faces • Their children are the boundaries of their holes**B**A C 0 A 1 2 3 1 2 3 B C 0 Inclusion tree**Linear Complex K**Exclusive and exhaustive collection of cells: vertices (0-cells), edges (1-cells), faces (2-cells), and in 3D volumes (3-cells). The boundary of any cell c of K is the union of cells of K. Each cell c of K can be asked for is set c.p, its dimension c.d, its boundary c.b, and its star c.s.**Test 1**• Let S=DiskLine. The series of figures, below, show the SGC, G, defined by S (i.e., S=G.p). From left to right, identify (fill in) the cells for SGCs yielding the following pointsets: S, S.e, S.b, S.i, S.k, and S.r. Make sure that you correctly classify the vertices (0-cells).**Test 2**• Consider the SGC, G, shown below left. Let S=G.p. In the subsequent figures, from left to right, mark (i.e. fill in) the cells of SGCs whose point sets are in the skin, wound, membrane, hair, cut, and dangle of S.